Where, the upper (lower) sign corresponds to bosonic (fermionic) contributions. The above integral \(J_{B,F}\)  can be expressed as a sum of the second kind modified Bessel functions \(K_{2} (x)\)  [1]},
\(J_{B,F}(y) = \lim _{N \rightarrow +\infty } \mp \sum _{l=1}^{N} {(\pm 1)^{l} y \over l^{2}} K_{2} (\sqrt{y} l)\;.\) 
